"Nasser M. Abbasi" schrieb:
>
> On 2/9/2024 11:23 AM,
clicl...@freenet.de wrote:
> >
> > I have been playing around with some old algebraic integrands in the
> > new version 1.3.10 of FriCAS on the web interface.
> >
> > Sam Blake's pseudo-elliptic of April 2020 still gives:
> >
> > integrate((x^4 - 1)*(x^4 + x^2 + 1)*sqrt(-x^4 + x^2 - 1)
> > /(x^4 + 1)^3, x)
> >
> >>> Error detected within library code:
> > catdef: division by zero
> >
> > perhaps because the radicand is negative everywhere on the real
> > axis.
> >
>
> Fyi;
>
> I've reported division by zero to Fricas newsgroup
>
>
https://groups.google.com/g/fricas-devel/c/6g0B53qX2TU
>
> Btw, I do not think many Fricas developers read sci.math.symbolic
>
> May be you could CC
>
>
fricas...@googlegroups.com
>
> also. I do not know if it will work or not from your end
> or if registration is needed or not. Sometimes I get direct email
> from the above myself.
Thanks, will try this next time and see what happpens. In fact, I more
or less regularly follow the posts at <
https://www.mail-archive.com/
fricas...@googlegroups.com>, and thus see Waldek occasionally
responding to <sci.math.symbolic> messages over there - so he's still
reading them. Dunno why he doesn't register at Eternal-September - he
even posted via Telekomunikacja Polska in April last year (I suppose
they offered a free trial which ran out).
I have verified that registration at <
www.solani.org> also works, but
one may have to remind the operators via e-mail and wait for a week
until one receives a password.
And as stated before, I can e-mail a password for <
news.killfile.org>
which I was able to guess to any one of the serious <sci.math.symbolic>
posters.
>
> > And an older and presumably truly elliptic case still fails:
> >
> > integrate((5*x - 9*sqrt(6) + 26)
> > /((x^2 - 4*x - 50)*sqrt(x^3 - 30*x - 56)), x)
> >
> >>> Error detected within library code:
> > catdef: division by zero
> >
> > in the same manner, although the radicand is cubic here.
> >
> > [...]
> >
> > And for the next integrand, FriCAS still produces unreasonable
> > integers in an arc tangent's argument:
> >
> > integrate(1/((x + 1)*(x^3 + 2)^(1/3)), x)
> >
> >
(log(((21*x^4+(-6)*x^3+(-96)*x^2+(-60)*x+12)*((x^3+2)^(1/3))^2+(21*x^5+(-48)*x^3+102*x^2+228*x+96)*(x^3+2)^(1/3)+(22*x^6+6*x^5+(-48)*x^4+44*x^3+24*x^2+(-192)*x+(-140)))/(x^6+6*x^5+15*x^4+20*x^3+15*x^2+6*x+1))+2*3^(1/2)*atan(((98966744593197647869364591874*x^4+190053406517364372745124029472*x^3+(-642339750020464731448133545632)*x^2+(-1764382450892402509391037276448)*x+(-1072244631963565627440642667696))*3^(1/2)*((x^3+2)^(1/3))^2+((-45228634350310035870300951616)*x^5+(-453545129950193664973324584892)*x^4+(-726175722499147186465445363320)*x^3+735314591615271415729365586328*x^2+2230842809300000322439227290544*x+1190118508012558386973005239952)*3^(1/2)*(x^3+2)^(1/3)+(93292570833559435663132301885*x^6+382151535711085278859235047618*x^5+673924074224408772959625384792*x^4+889426563183087468015580290048*x^3+888876515195959220955879945824*x^2+351260598258508240019971964880*x+(-47674000995597211057816884304))*3^(1/2))/(236716304443694165237125394649*x^6+1013240117509374668590043803350*x^5+46796858328175763683008212928*x^4+(-2686291575945300326054363894472)*x^3+1085003586721431086608600126056*x^2+7625406903034897531937916271008*x+4664445860470002276943457906640)))/12
> >
> > ... while the antiderivative can in fact be compactly stated as:
> >
> > INT(1/((x + 1)*(x^3 + 2)^(1/3)), x) =
> > 1/12*(- 3*LN((x^3 + 2)^(1/3) - x)
> > + 2*SQRT(3)*ATAN(1/SQRT(3)*(1 + 2*x/(x^3 + 2)^(1/3))))
> > - 1/4*(LN((x + 2)^3 - (x^3 + 2))
> > - 3*LN((x + 2) - (x^3 + 2)^(1/3))
> > + 2*SQRT(3)*ATAN(1/SQRT(3)*(1 + 2*((x + 2)/(x^3 + 2)^(1/3)))))
> >
> > If the unreasonable numbers cannot be avoided earlier, they could at
> > least be removed by subtracting an arc tangent for a suitably chosen
> > value of x; both x = infinity and x = -2^(1/3) turn out to work
> > well.
I also find that x = -1 works less well; perhaps one should simply try
x = infinity in all cases of algebraic antiderivatives with
unreasonable arc tangent arguments (but only if the radical stays
real?), and perhaps for reasonable arguments as well to avoid deciding
what's unreasonable.
> >
> > [...]
> >
Martin.